Extending Goldberg's method to parametrize and control the geometry of Goldberg polyhedra.

Goldberg polyhedra have been widely studied across multiple fields, as their distinctive pattern can lead to many useful applications. Their topology can be determined using Goldberg's method through generating topologically equivalent structures, named cages. However, the geometry of Goldberg polyhedra remains underexplored. This study extends Goldberg's framework to a new method that can systematically determine the topology and effectively control the geometry of Goldberg polyhedra based on the initial shapes of cages. In detail, we first parametrize the cage's geometry under specified topology and polyhedral symmetry; then, we manipulate the predefined independent variables through optimization to achieve the user-defined geometric properties. The benchmark problem of finding equilateral Goldberg polyhedra is solved to demonstrate the effectiveness of the proposed method. Using this method, we have successfully achieved nearly exact spherical Goldberg polyhedra, with all vertices on a sphere and all faces being planar under extremely low numerical errors. Such results serve as strong numerical evidence for the existence of this new type of Goldberg polyhedra. Furthermore, we iteratively perform k-means clustering and optimization to significantly reduce the number of different edge lengths to benefit the cost reduction for architectural and engineering applications.